Explicit error bounds in a conforming finite element method
Mathematics of Computation
A unified approach to a posteriori error estimation based on duality error majorants
Mathematics and Computers in Simulation - Special issue from IMACS sponsored conference: “Modelling '98”
A posteriori error estimation for variational problems with uniformly convex functionals
Mathematics of Computation
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Journal of Scientific Computing
An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method
Journal of Scientific Computing
Short Note: An explicit expression for the penalty parameter of the interior penalty method
Journal of Computational Physics
Understanding And Implementing the Finite Element Method
Understanding And Implementing the Finite Element Method
Estimation of penalty parameters for symmetric interior penalty Galerkin methods
Journal of Computational and Applied Mathematics
A posteriori error estimators for locally conservative methods of nonlinear elliptic problems
Applied Numerical Mathematics
A Posteriori Error Estimation for Discontinuous Galerkin Finite Element Approximation
SIAM Journal on Numerical Analysis
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We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.